Termination criteria for the one-dimensional interval version of newton&#39;s method

ABSTRACT

One embodiment of the present invention provides a system for finding zeros of a function, f, within an interval, X, using the interval version of Newton&#39;s method. The system operates by receiving a representation of the interval X. This representation including a first floating-point number, a, representing the left endpoint of X, and a second floating-point number, b, representing the right endpoint of X. Next, the system performs an interval Newton step on X, wherein the point of expansion is the midpoint, x, of the interval X. Note that performing the interval Newton step involves evaluating f(x) to produce an interval result f I (x). If f I (x) contains zero, the system evaluates f(a) to produce an interval result f I (a). It also evaluates f(b) to produce an interval result f I (b). The system then evaluates a termination condition for the processing of the current interval X, wherein the termination condition is TRUE if a number of sub-conditions are satisfied, including if f I (a) contains zero and if f I (b) contains zero. If the termination condition is TRUE, the system terminates the processing of the current interval X, and records X as a final bound.

BACKGROUND

[0001] 1. Field of the Invention

[0002] The present invention relates to performing arithmetic operationson interval operands within a computer system. More specifically, thepresent invention relates to a method and an apparatus for finding theroots of a nonlinear equation of one variable using the interval versionof Newton's method.

[0003] 2. Related Art

[0004] Rapid advances in computing technology make it possible toperform trillions of computational operations each second. Thistremendous computational speed makes it practical to performcomputationally intensive tasks as diverse as predicting the weather andoptimizing the design of an aircraft engine. Such computational tasksare typically performed using machine-representable floating-pointnumbers to approximate values of real numbers. (For example, see theInstitute of Electrical and Electronics Engineers (IEEE) standard 754for binary floating-point numbers.)

[0005] In spite of their limitations, floating-point numbers aregenerally used to perform most computational tasks.

[0006] One limitation is that machine-representable floating-pointnumbers have a fixed-size word length, which limits their accuracy. Notethat a floating-point number is typically encoded using a 32, 64 or128-bit binary number, which means that there are only 2³², 2⁶⁴ or 2¹²⁸possible symbols that can be used to specify a floating-point number.Hence, most real number values can only be approximated with acorresponding floating-point number. This creates estimation errors thatcan be magnified through even a few computations, thereby adverselyaffecting the accuracy of a computation.

[0007] A related limitation is that floating-point numbers contain noinformation about their accuracy. Most measured data values include someamount of error that arises from the measurement process itself. Thiserror can often be quantified as an accuracy parameter, which cansubsequently be used to determine the accuracy of a computation.However, floating-point numbers are not designed to keep track ofaccuracy information, whether from input data measurement errors ormachine rounding errors. Hence, it is not possible to determine theaccuracy of a computation by merely examining the floating-point numberthat results from the computation.

[0008] Interval arithmetic has been developed to solve theabove-described problems. Interval arithmetic represents numbers asintervals specified by a first (left) endpoint and a second (right)endpoint. For example, the interval [a, b], where a<b, is a closed,bounded subset of the real numbers, R, which includes a and b as well asall real numbers between a and b. Arithmetic operations on intervaloperands (interval arithmetic) are defined so that interval resultsalways contain the entire set of possible values. The result is amathematical system for rigorously bounding numerical errors from allsources, including measurement data errors, machine rounding errors andtheir interactions. (Note that the first endpoint normally contains the“infimum”, which is the largest number that is less than or equal toeach of a given set of real numbers. Similarly, the second endpointnormally contains the “supremum”, which is the smallest number that isgreater than or equal to each of the given set of real numbers.)

[0009] One commonly performed computational operation is to find theroots of a nonlinear equation using Newton's method. The intervalversion of Newton's method works in the following manner. From the meanvalue theorem,

f(x)−f(x*)=(x−x*)f′(ξ),

[0010] where ξ is some generally unknown point between x and x*. If x*is a zero of f, then f(x*)=0 and, from the previous equation,

x*=x−f(x)/f′(ξ).

[0011] Let X be an interval containing both x and x*. Since ξ is betweenx and x*, it follows that ξεX Moreover, from basic properties ofinterval analysis it follows that f′(ξ) εf′(X). Hence, x * εN (x,X)where

N(x,X)=x−f(x)/f′(X).

[0012] Temporarily assume 0 ∉f′(A) so that N(x,A) is a finite interval.Since any zero of f in X is also in N(x,X), the zero is in theintersection X ∩N(x,X). Using this fact, we define an algorithm forfinding zero x*. Let X₀ be an interval containing x*. For n=0, 1, 2, . .. , define

X_(n)=m(X_(n))

N(x _(n) ,X _(n))=x _(n) −f(x _(n))/f′(X _(n))

X _(n=1) =X _(n) ∩N(x _(n) ,X _(n)),

[0013] wherein m(X) is the midpoint of the interval X. We call x_(n) thepoint of expansion for the Newton method. It is not necessary to choosex_(n) to be the midpoint of X_(n). The only requirement is thatx_(n)εX_(n) to assure that x*εN(x_(n),X_(n)). However, it is convenientand efficient to choose x_(n)=m(X_(n)).

[0014] Roots of an interval equation can be intervals rather than pointswhen the equation contains non-degenerate interval constants orparameters. Suppose the interval version of Newton's method to find theroots of a single nonlinear equation has not yet satisfied theuser-specified convergence tolerances. Then it is difficult todistinguish between the following three situations:

[0015] a) the current interval is a tight enclosure of a single intervalroot;

[0016] b) the current interval contains sufficiently distinct intervalroots that they can be isolated with a reasonable amount of effort; and

[0017] c) the current interval contains point and/or interval roots thatare so close to being indistinguishable that the effort to isolate themis unreasonably large with the existing wordlength.

[0018] What is needed is a method and an apparatus for terminating theinterval version of Newton's root finding method before iterations losetheir practical value in isolating meaningfully distinct interval roots.

SUMMARY

[0019] One embodiment of the present invention provides a system forfinding zeros of a function, f within an interval, X, using the intervalversion of Newton's method. The system operates by receiving arepresentation of the interval X. This representation including a firstfloating-point number, a, representing the left endpoint of X, and asecond floating-point number, b, representing the right endpoint of X.Next, the system performs an interval Newton step on X, wherein thepoint of expansion is the midpoint, x, of the interval X. Note thatperforming the interval Newton step involves evaluating f(x) to producean interval result f^(I)(x). If f^(I)(x) contains zero, the systemevaluates f(a) to produce an interval result f^(I)(a). It also evaluatesf(b) to produce an interval result f^(I)(b). The system then evaluates atermination condition for the processing of the current interval X,wherein the termination condition is TRUE if a number of sub-conditionsare satisfied, including if f^(I)(a) contains zero and if f^(I)(b)contains zero. If the termination condition is TRUE, the systemterminates the processing of the current interval X, and records X as afinal bound.

[0020] In one embodiment of the present invention, if f^(I)(a) does notcontain zero, evaluating f(a) additionally involves performing aninterval Newton step wherein the point of expansion is a.

[0021] In one embodiment of the present invention, if f^(I)(b) does notcontain zero, evaluating f(b) additionally involves performing aninterval Newton step wherein the point of expansion is b.

[0022] In one embodiment of the present invention, if f^(I)(x) containszero and if f′(X) contains zero, the termination condition forprocessing the current interval X is TRUE if f^(I)(a) contains zero,f^(I)(b) contains zero, f^(I)(x₁) contains zero and if f^(I)(x₂)contains zero. Note that x₁ is the midpoint between a and x; and x₂ isthe midpoint between x and b.

[0023] In one embodiment of the present invention, if f^(I)(x) containszero and if f′(X) contains zero, and if either f^(I)(x₁) or f^(I)(x₂)does not contain zero, the system additionally splits the interval X inhalf and applies the interval Newton method to each half separately.

[0024] In one embodiment of the present invention, in the case wheref^(I)(x) does not contain zero and if f′(X) contains zero, if the widthof the interval X divided by the magnitude of the interval X is lessthan a first threshold value, and the magnitude of f(X) is less than asecond threshold value, the method further comprises terminating theprocessing of the current interval X, and recording X as a final bound.

[0025] In one embodiment of the present invention, if a given Newtonstep does not reduce the width of an interval by at least half, thesystem additionally splits the interval in half and applies the intervalNewton method to each half separately.

[0026] In one embodiment of the present invention, if an interval Newtonstep results in two intervals, the system additionally applies theinterval Newton method to each of the two intervals separately.

[0027] In one embodiment of the present invention, if the result of aninterval Newton step is the empty interval, the system returns toprocess another interval.

[0028] In one embodiment of the present invention, if f^(I)(a) containszero and if f^(I)(b) contains zero, the system returns to processanother interval.

[0029] In one embodiment of the present invention, if f^(I)(a) does notcontain zero or if f^(I)(b) does not contain zero, the system performsan interval Newton step wherein the point of expansion is the midpointof the interval X. If the result of this interval Newton step is theempty interval, the system returns to process another interval.

[0030] In one embodiment of the present invention, the system terminatesthe processing of the current interval X after a predetermined number ofiterations.

BRIEF DESCRIPTION OF THE FIGURES

[0031]FIG. 1 illustrates a computer system in accordance with anembodiment of the present invention.

[0032]FIG. 2 illustrates the process of compiling and using code forinterval computations in accordance with an embodiment of the presentinvention.

[0033]FIG. 3 illustrates an arithmetic unit for interval computations inaccordance with an embodiment of the present invention.

[0034]FIG. 4 is a flow chart illustrating the process of performing aninterval computation in accordance with an embodiment of the presentinvention.

[0035]FIG. 5 illustrates four different interval operations inaccordance with an embodiment of the present invention.

[0036]FIG. 6 illustrates the process of solving for zeros of a functionusing the interval Newton method in accordance with an embodiment of thepresent invention.

[0037]FIG. 7 illustrates the interval Newton method for the case wherethe derivative of the function does not contain zero in accordance withan embodiment of the present invention.

[0038]FIG. 8 illustrates the interval Newton method for the case wherethe derivative of the function contains zero in accordance with anembodiment of the present invention.

DETAILED DESCRIPTION

[0039] The following description is presented to enable any personskilled in the art to make and use the invention, and is provided in thecontext of a particular application and its requirements. Variousmodifications to the disclosed embodiments will be readily apparent tothose skilled in the art, and the general principles defined herein maybe applied to other embodiments and applications without departing fromthe spirit and scope of the present invention. Thus, the presentinvention is not intended to be limited to the embodiments shown, but isto be accorded the widest scope consistent with the principles andfeatures disclosed herein.

[0040] The data structures and code described in this detaileddescription are typically stored on a computer readable storage medium,which may be any device or medium that can store code and/or data foruse by a computer system. This includes, but is not limited to, magneticand optical storage devices such as disk drives, magnetic tape, CDs(compact discs) and DVDs (digital versatile discs or digital videodiscs), and computer instruction signals embodied in a transmissionmedium (with or without a carrier wave upon which the signals aremodulated). For example, the transmission medium may include acommunications network, such as the Internet.

[0041] Computer System

[0042]FIG. 1 illustrates a computer system 100 in accordance with anembodiment of the present invention. As illustrated in FIG. 1, computersystem 100 includes processor 102, which is coupled to a memory 112 anda to peripheral bus 110 through bridge 106. Bridge 106 can generallyinclude any type of circuitry for coupling components of computer system100 together.

[0043] Processor 102 can include any type of processor, including, butnot limited to, a microprocessor, a mainframe computer, a digital signalprocessor, a personal organizer, a device controller and a computationalengine within an appliance. Processor 102 includes an arithmetic unit104, which is capable of performing computational operations usingfloating-point numbers.

[0044] Processor 102 communicates with storage device 108 through bridge106 and peripheral bus 110. Storage device 108 can include any type ofnon-volatile storage device that can be coupled to a computer system.This includes, but is not limited to, magnetic, optical, andmagneto-optical storage devices, as well as storage devices based onflash memory and/or battery-backed up memory.

[0045] Processor 102 communicates with memory 112 through bridge 106.Memory 112 can include any type of memory that can store code and datafor execution by processor 102. As illustrated in FIG. 1, memory 112contains computational code for intervals 114. Computational code 114contains instructions for the interval operations to be performed onindividual operands, or interval values 115, which are also storedwithin memory 112. This computational code 114 and these interval values115 are described in more detail below with reference to FIGS. 2-5.

[0046] Note that although the present invention is described in thecontext of computer system 100 illustrated in FIG. 1, the presentinvention can generally operate on any type of computing device that canperform computations involving floating-point numbers. Hence, thepresent invention is not limited to the computer system 100 illustratedin FIG. 1.

[0047] Compiling and Using Interval Code

[0048]FIG. 2 illustrates the process of compiling and using code forinterval computations in accordance with an embodiment of the presentinvention. The system starts with source code 202, which specifies anumber of computational operations involving intervals. Source code 202passes through compiler 204, which converts source code 202 intoexecutable code form 206 for interval computations. Processor 102retrieves executable code 206 and uses it to control the operation ofarithmetic unit 104.

[0049] Processor 102 also retrieves interval values 115 from memory 112and passes these interval values 115 through arithmetic unit 104 toproduce results 212. Results 212 can also include interval values.

[0050] Note that the term “compilation” as used in this specification isto be construed broadly to include pre-compilation and just-in-timecompilation, as well as use of an interpreter that interpretsinstructions at run-time. Hence, the term “compiler” as used in thespecification and the claims refers to pre-compilers, just-in-timecompilers and interpreters.

[0051] Arithmetic Unit for Intervals

[0052]FIG. 3 illustrates arithmetic unit 104 for interval computationsin more detail accordance with an embodiment of the present invention.Details regarding the construction of such an arithmetic unit are wellknown in the art. For example, see U.S. Pat. Nos. 5,687,106 and6,044,454, which are hereby incorporated by reference in order toprovide details on the construction of such an arithmetic unit.Arithmetic unit 104 receives intervals 302 and 312 as inputs andproduces interval 322 as an output.

[0053] In the embodiment illustrated in FIG. 3, interval 302 includes afirst floating-point number 304 representing a first endpoint ofinterval 302, and a second floating-point number 306 representing asecond endpoint of interval 302. Similarly, interval 312 includes afirst floating-point number 314 representing a first endpoint ofinterval 312, and a second floating-point number 316 representing asecond endpoint of interval 312. Also, the resulting interval 322includes a first floating-point number 324 representing a first endpointof interval 322, and a second floating-point number 326 representing asecond endpoint of interval 322.

[0054] Note that arithmetic unit 104 includes circuitry for performingthe interval operations that are outlined in FIG. 5. This circuitryenables the interval operations to be performed efficiently.

[0055] However, note that the present invention can also be applied tocomputing devices that do not include special-purpose hardware forperforming interval operations. In such computing devices, compiler 204converts interval operations into a executable code that can be executedusing standard computational hardware that is not specially designed forinterval operations.

[0056]FIG. 4 is a flow chart illustrating the process of performing aninterval computation in accordance with an embodiment of the presentinvention. The system starts by receiving a representation of aninterval, such as first floating-point number 304 and secondfloating-point number 306 (step 402). Next, the system performs anarithmetic operation using the representation of the interval to producea result (step 404). The possibilities for this arithmetic operation aredescribed in more detail below with reference to FIG. 5.

[0057] Interval Operations

[0058]FIG. 5 illustrates four different interval operations inaccordance with an embodiment of the present invention. These intervaloperations operate on the intervals X and Y. The interval X includes twoendpoints,

[0059] x denotes the lower bound of X, and

[0060] x denotes the upper bound of X.

[0061] The interval X is a closed subset of the extended (including −∞and +∞) real numbers R * (see line 1 of FIG. 5). Similarly the intervalY also has two endpoints and is a closed subset of the extended realnumbers R * (see line 2 of FIG. 5).

[0062] Note that an interval is a point or degenerate interval if X=[x,x]. Also note that the left endpoint of an interior interval is alwaysless than or equal to the right endpoint. The set of extended realnumbers, R * is the set of real numbers, R, extended with the two idealpoints negative infinity and positive infinity:

R*=R ∪{−∞}∪{+∞}.

[0063] In the equations that appear in FIG. 5, the up arrows and downarrows indicate the direction of rounding in the next and subsequentoperations. Directed rounding (up or down) is applied if the result of afloating-point operation is not machine-representable.

[0064] The addition operation X+Y adds the left endpoint of X to theleft endpoint of Y and rounds down to the nearest floating-point numberto produce a resulting left endpoint, and adds the right endpoint of Xto the right endpoint of Y and rounds up to the nearest floating-pointnumber to produce a resulting right endpoint.

[0065] Similarly, the subtraction operation X−Y subtracts the rightendpoint of Y from the left endpoint of X and rounds down to produce aresulting left endpoint, and subtracts the left endpoint of Y from theright endpoint of X and rounds up to produce a resulting right endpoint.

[0066] The multiplication operation selects the minimum value of fourdifferent terms (rounded down) to produce the resulting left endpoint.These terms are: the left endpoint of X multiplied by the left endpointof Y; the left endpoint of X multiplied by the right endpoint of Y; theright endpoint of X multiplied by the left endpoint of Y; and the rightendpoint of X multiplied by the right endpoint of Y. This multiplicationoperation additionally selects the maximum of the same four terms(rounded up) to produce the resulting right endpoint.

[0067] Similarly, the division operation selects the minimum of fourdifferent terms (rounded down) to produce the resulting left endpoint.These terms are: the left endpoint of X divided by the left endpoint ofY; the left endpoint of X divided by the right endpoint of Y; the rightendpoint of X divided by the left endpoint of Y; and the right endpointof X divided by the right endpoint of Y. This division operationadditionally selects the maximum of the same four terms (rounded up) toproduce the resulting right endpoint. For the special case where theinterval Y includes zero, X/Y is an exterior interval that isnevertheless contained in the interval R*.

[0068] Note that the result of any of these interval operations is theempty interval if either of the intervals, X or Y, are the emptyinterval. Also note, that in one embodiment of the present invention,extended interval operations never cause undefined outcomes, which arereferred to as “exceptions” in the IEEE 754 standard.

[0069] Interval Version of Newton's Method

[0070]FIG. 6 illustrates the process of solving for zeros of a functionusing the interval version of Newton's method in accordance with anembodiment of the present invention. The system starts with an initialinterval X₀ and stopping tolerances ε_(X) and ε_(F). The system stopswhen interval bounds for all zeros of f in X₀ have been found. Aftertermination, the bounds on any simple zero generally approximate theoptimal bound. If the tolerances ε_(X) and ε_(F) are not chosen toosmall, then each multiple zero of f in X₀ is generally isolated withinan interval of relative width less than ε_(X). Also, we generally have|f(x)|<ε_(F) for all points x in an interval bounding a zero.

[0071] In the following description, the current interval is denoted byX at each step although X changes from step to step. Also, x denotes themidpoint m(X) so x changes as well. Note that except for a few casesdescribed below, the Newton step is generally defined by an expansionabout x.

[0072] The system starts by placing the initial interval, X₀, in a listof intervals to be processed, L (step 602).

[0073] If L is empty, the system stops. Otherwise, the system selectsthe interval X, from L that has been in L for the shortest amount oftime, and the system deletes X from L (step 604).

[0074] Next, the system determines if the derivative off which isrepresented as f′, when evaluated over the interval X contains zero. Inother words, is 0 εf′(X)? (step 606). If not, the system finds roots forthe case where f′(X) does not contain zero (step 608). The operationsinvolved in solving for this case are described in more detail withreference to FIG. 7 below. The system then returns to step 604 processadditional intervals if necessary.

[0075] If f′(X) contains zero at step 606, the system determines whetherf^(I)(x) contains zero (step 610). Note that f^(I)(x) denotes theinterval resulting from an evaluation of f(x). If the conditions in bothsteps 606 and 610 are true, the system finds the roots of f for the casewhere f′(X) does not contain zero (step 612). The operations involved insolving for this case are described in more detail with reference toFIG. 8 below. The system then returns to step 604 to process additionalintervals if necessary.

[0076] If f^(I)(x) does not contain zero at step 610, the systemperforms tests involving the stopping tolerances ε_(X) and ε_(F) (step614). If the width of the interval X(denoted as w(X)) divided by themagnitude of X (denoted as |X|) is less than ε_(X), and if the magnitudeof f(X) (denoted as |f(X)|) is less than ε_(F), the system records X asa final bound (step 616) and returns to step 604 to process additionalintervals if necessary. Note that the width of the interval X=[a,b](denoted as w(X)) is simply b−a. Also note that the magnitude of aninterval X=[a, b] (denoted by |X|) is the maximum value of |x| for all xε X. Thus, |X|=max(|a|,|b|). Furthermore, note that the stoppingtolerances, ε_(X) and β_(F), are used to stop the system from performingfurther computations when the interval bound on a zero of f issufficiently narrow.

[0077] If at step 614 one or both of the tested quantities is greaterthan or equal to the corresponding stopping tolerance, the systemapplies the interval Newton step to the interval X (step 617).

[0078] The system then examines the result of the Newton step (step618). If the result is empty, the system returns to step 604 to processadditional intervals, if necessary. If the result is a single interval,the system determines whether the Newton step reduced the interval X byat least half in size (step 620). If so, the system returns to step 606to continue processing the current interval X. Otherwise, the systemsplits X into halves, puts one half into the list of intervals to beprocessed, L, and designates the other half as the current interval X(step 622). The system then returns to step 606 to continue processingthe current interval X.

[0079] If the result of the Newton step at step 618 is two intervals,the system puts one of the two intervals into the list of intervals tobe processed, L, and designates the other interval as the currentinterval, X (step 624). The system then returns to step 606 to continueprocessing the current interval X.

[0080] Case Where 0 ∉ f′(X)

[0081]FIG. 7 illustrates the interval Newton method for the case wherethe derivative of the function does not contain zero in accordance withan embodiment of the present invention. This flow chart illustrates inmore detail the operations carried out at step 608 of FIG. 6.

[0082] Note that the relation 0 ∉ f′(X) assures that there is no morethan one point or interval zero of f in X. Moreover, if there is a zeroof f in X, then it is a simple one. Also note that inclusion isotonicityassures that 0 ∉ f′(X′) for any interval X′ contained in X.

[0083] The system starts by initializing a counter variable, n, and twoflags, F_(a) and F_(b), to zero (step 702). Note that the system setsF_(a)=1 when 0 ∉ f′(a). Similarly, the system sets F_(b)=1 when 0 ∉f′(b). Also note that the system cycles through the operations listedbelow no more than four times. The integer n counts the cycles.

[0084] After initializing the variables in step 702, the system testsF_(a) to see if F_(a)==1 (step 703). If F_(a)==1, the system proceeds tostep 710 to test flag F_(b). On the other hand, if F_(a)≠1 at step 703,the system evaluates f(a) to produce the resulting interval f^(I)(a)(step 704). Next, the system determines if f^(I)(a) contains zero (step706). If so, the system sets F_(a)=1 (step 707) and proceeds to step 710to test flag F_(b). If at step 706 f^(I)(a) does not contain zero, thesystem applies an interval Newton step wherein the point of expansion is“a” (step 708). If the result of this expansion is the empty interval,the system returns to process the next interval.

[0085] Otherwise, the system tests F_(b) to see if F_(b)==1 (step 710).If F_(b)==1, the system proceeds to step 718 to test both flags F_(a)and F_(b). On the other hand, if F_(b)≠1 at step 710, the systemevaluates f(b) to produce the resulting intervalr f^(I)(b) (step 712).Next, the system determines if f^(I)(b) contains zero (step 714). If so,the system sets F_(b)=1 (step 715) and proceeds to step 718 to test bothflags F_(a) and F_(b). If at step 714 f^(I)(b) does not contain zero,the system applies an interval Newton step wherein the point ofexpansion is “b” (step 716). If the result of this expansion is theempty interval, the system returns to process the next interval. Let x**be the midpoint of X that satisfies the condition 0 68 f^(I)(x) in step610. x** is set in step 612. Note that if x** is no longer an element ofX, the system uses the midpoint of the current X, m(X), as the point ofexpansion.

[0086] Otherwise, the system tests to see if both F_(a)==1 and F_(b)==1(step 718). If both F_(a)==1 and F_(b)==1, the bounds, a and b, on thezero of f are as tight as they are likely to become. Hence, the systemreturns to process the next interval (step 721).

[0087] Otherwise, the system applies an interval Newton step wherein thepoint of expansion is my (step 720). If the result of this expansion isthe empty interval, the system returns to process the next interval.Next, the system increments n (step 722) and tests to see if n<4 (step724). If n≧4, the system returns to process the next interval becausethe bounds on the zero are unlikely to be further tightened byadditional processing (step 726). Otherwise, if n<4, the system returnsto step 703 to repeat the process.

[0088] Case Where 0 ε f′(X)

[0089]FIG. 8 illustrates the interval Newton method for the case wherethe derivative of the function contains zero and 0 εf^(I)(x) inaccordance with an embodiment of the present invention. This flow chartillustrates in more detail the operations carried out at step 612 ofFIG. 6.

[0090] The system first evaluates f(a) to produce the resulting intervalf^(I)(a) (step 802). Next, the system determines if f^(I)(a) containszero (step 804). If not, the system applies an interval Newton stepwherein the point of expansion is “a” (step 806) and returns to the mainprogram.

[0091] On the other hand, if f^(I)(a) contains zero, the systemevaluates f(b) to produce the resulting interval f^(I)(b) (step 808).Next, the system determines if f^(I)(b) contains zero (step 810). Ifnot, the system applies an interval Newton step wherein the point ofexpansion is “b” (step 812) and returns to the main program.

[0092] On the other hand, if f^(I)(b) contains zero, the system nextfinds the centers, x₁ and x₂, of the two halves of the interval X bycomputing x₁=(3a+b)/4 and X₂=(a+3b)/4 (step 814). The system thenevaluates the functions f at each of the centers, x₁ and x₂, to producethe resulting intervals f^(I)(x₁) and f^(I)(x₂). Next, the systemdetermines if both f(x₁) contains zero and f(x₂) contains zero (step816). If so, the computed (interval) value of f contains zero for eachof the five equally spaced points, a, x₁, m(X), x₂ and b. In this case,additional processing is unlikely to narrow the X further. Hence, thesystem accepts X as a final bound on a zero of f (step 822).

[0093] On the other hand, if either f^(I)(x₁) or f^(I)(x₂) does notcontain zero, the system splits the interval X in half, and applies theinterval Newton method to each half separately (step 820).

[0094] Note that the system can also make use of higher precisioninterval arithmetic to help resolve the uncertainty when 0 ε f^(I)(x)and 0 ε f′(X).

[0095] The foregoing descriptions of embodiments of the presentinvention have been presented for purposes of illustration anddescription only. They are not intended to be exhaustive or to limit thepresent invention to the forms disclosed. Accordingly, manymodifications and variations will be apparent to practitioners skilledin the art.

[0096] For example, in the computational process described above withreference to FIG. 6, it is possible to process intervals in the list, L,in parallel using multiple concurrently executing processors.Furthermore, it is possible to subdivide the initial interval intomultiple smaller sub-intervals so that the multiple sub-intervals can beprocessed in parallel.

[0097] In another embodiment of the present invention, if the function,f, has already been evaluated at a right endpoint or a left endpoint ofan interval, X, the system saves the result of this evaluation in thelist, L, so that the function does not have to be reevaluated at theright endpoint or left endpoint of X.

[0098] Hence, the above disclosure is not intended to limit the presentinvention. The scope of the present invention is defined by the appendedclaims.

What is claimed is:
 1. A method for finding zeros of a function, f,within an interval, X, using the interval version of Newton's method,wherein f′ is the derivative of the function f, the method comprising:receiving a representation of the interval X, the representationincluding a first floating-point number, a, representing the leftendpoint of X, and a second floating-point number, b, representing theright endpoint of X; performing an interval Newton step on X, whereinthe point of expansion is the midpoint, x, of the interval X, andwherein performing the interval Newton step involves evaluating f(x) toproduce an interval result f^(I)(x); and if f^(I)(x) contains zero,evaluating f^(I)(a) to produce an interval result f^(I)(a), evaluatingf(b) to produce an interval result f^(I)(b), evaluating a terminationcondition for the processing of the current interval X, wherein thetermination condition is TRUE if a number of conditions are satisfied,including if f^(I)(a) contains zero and if f^(I)(b) contains zero, andif the termination condition is TRUE, terminating the processing of thecurrent interval X, and recording X as a final bound.
 2. The method ofclaim 1, wherein if f^(I)(a) does not contain zero, evaluating f(a)additionally involves performing an interval Newton step wherein thepoint of expansion is a.
 3. The method of claim 1, wherein if f^(I)(b)does not contain zero, evaluating f(b) additionally involves performingan interval Newton step wherein the point of expansion is b.
 4. Themethod of claim 1, wherein if f^(I)(x) contains zero and f′(X) containszero, the termination condition for processing the current interval X isTRUE if f^(I)(a) contains zero, f^(I)(b) contains zero, f^(I)(x₁)contains zero and f^(I)(x₂) contains zero; wherein x₁ is the midpointbetween a and x; and wherein x₂ is the midpoint between x and b.
 5. Themethod of claim 4, wherein if f^(I)(x) contains zero and if f′(X)contains zero, and if either f^(I)(x₁) or f^(I)(x₂) does not containzero, the method further comprises: splitting the interval X in half;and applying the interval Newton method to each half separately.
 6. Themethod of claim 1, wherein if f^(I)(x) does not contain zero and iff′(X) contains zero, the method further comprises terminating theprocessing of the current interval X, and recording X as a final boundon condition that: the width of the interval X divided by the magnitudeof the interval X is less than a first threshold value; and themagnitude of f(A) is less than a second threshold value.
 7. The methodof claim 1, wherein if a given Newton step does not reduce the width ofan interval by at least half, the method further comprises splitting theinterval in half and applying the interval Newton method to each of thetwo halves separately.
 8. The method if claim 1, wherein if an intervalNewton step results in two intervals, the method further comprisesapplying the interval Newton method to each of the two intervalsseparately.
 9. The method of claim 1, wherein if the result of aninterval Newton step is the empty interval, the method returns toprocess another interval.
 10. The method of claim 1, wherein if f^(I)(a)contains zero and if f^(I)(b) contains zero, the method returns toprocess another interval.
 11. The method of claim 1, wherein if f^(I)(a)does not contain zero or if f^(I)(b) does not contain zero, the methodfurther comprises performing an interval Newton step wherein the pointof expansion is the midpoint of the interval X; and wherein if theresult of the interval Newton step about the midpoint is the emptyinterval, the method returns to process another interval.
 12. The methodof claim 11, further comprising terminating the processing of thecurrent interval X after a predetermined number of iterations.
 13. Acomputer-readable storage medium storing instructions that when executedby a computer cause the computer to perform a method for finding zerosof a function, f within an interval, X, using the interval version ofNewton's method, wherein f′ is the derivative of the function f, themethod comprising: receiving a representation of the interval X, therepresentation including a first floating-point number, a, representingthe left endpoint of X, and a second floating-point number, b,representing the right endpoint of X; performing an interval Newton stepon X, wherein the point of expansion is the midpoint, x, of the intervalX, and wherein performing the interval Newton step involves evaluatingf(x) to produce an interval result f^(I)(x); and if f^(I)(x) containszero, evaluating f(a) to produce an interval result f^(I)(a), evaluatingf(b) to produce an interval result f^(I)(b), evaluating a terminationcondition for the processing of the current interval X, wherein thetermination condition is TRUE if a number of conditions are satisfied,including if f^(I)(a) contains zero and if f^(I)(b) contains zero, andif the termination condition is TRUE, terminating the processing of thecurrent interval X, and recording X as a final bound.
 14. Thecomputer-readable storage medium of claim 13, wherein if f^(I)(a) doesnot contain zero evaluating f(a) additionally involves performing aninterval Newton step wherein the point of expansion is a.
 15. Thecomputer-readable storage medium of claim 13, wherein if f^(I)(b) doesnot contain zero, evaluating f(b) additionally involves performing aninterval Newton step wherein the point of expansion is b.
 16. Thecomputer-readable storage medium of claim 13, wherein if f^(I)(x)contains zero and f′(X) contains zero, the termination condition forprocessing the current interval X is TRUE if f^(I)(a) contains zero,f^(I)(b) contains zero, f^(I)(x₁) contains zero and f^(I)(x₂) containszero; wherein x₁ is the midpoint between a and x; and wherein x₂ is themidpoint between x and b.
 17. The computer-readable storage medium ofclaim 16, wherein if f^(I)(x) contains zero and if f′(X) contains zero,and if either f^(I)(x₁) or f^(I)(x₂) does not contain zero, the methodfurther comprises: splitting the interval X in half; and applying theinterval Newton method to each half separately.
 18. Thecomputer-readable storage medium of claim 13, wherein if f^(I)(x) doesnot contain zero and if f′(X) contains zero, the method furthercomprises terminating the processing of the current interval X, andrecording X as a final bound on condition that: the width of theinterval X divided by the magnitude of the interval X is less than afirst threshold value; and the magnitude of f(X) is less than a secondthreshold value.
 19. The computer-readable storage medium of claim 13,wherein if a given Newton step does not reduce the width of an intervalby at least half, the method further comprises splitting the interval inhalf and applying the interval Newton method to each of the two halvesseparately.
 20. The computer-readable storage medium if claim 13,wherein if an interval Newton step results in two intervals, the methodfurther comprises applying the interval Newton method to each of the twointervals separately.
 21. The computer-readable storage medium of claim13, wherein if the result of an interval Newton step is the emptyinterval, the method returns to process another interval.
 22. Thecomputer-readable storage medium of claim 13, wherein if f^(I)(a)contains zero and if f^(I)(b) contains zero, the method returns toprocess another interval.
 23. The computer-readable storage medium ofclaim 13, wherein if f^(I)(a) does not contain zero or if f^(I)(b) doesnot contain zero, the method further comprises performing an intervalNewton step wherein the point of expansion is the midpoint of theinterval X; and wherein if the result of the interval Newton step aboutthe midpoint is the empty interval, the method returns to processanother interval.
 24. The computer-readable storage medium of claim 13,wherein the method further comprises terminating the processing of thecurrent interval X after a predetermined number of iterations.
 25. Anapparatus that finds zeros of a function, f within an interval, X, usingthe interval version of Newton's method, wherein f′ is the derivative ofthe function f the apparatus comprising: a receiving mechanism that isconfigured to receive a representation of the interval X, therepresentation including a first floating-point number, a, representingthe left endpoint of X, and a second floating-point number, b,representing the right endpoint of X, an interval Newton mechanism thatis configured to perform an interval Newton step on X, wherein the pointof expansion is the midpoint, x, of the interval X, and whereinperforming the interval Newton step involves evaluating f(x) to producean interval result f^(I)(x); and wherein if f^(I)(x) contains zero, theinterval Newton mechanism is configured to, evaluate f(a) to produce aninterval result f^(I)(a), evaluate f(b) to produce an interval resultf^(I)(b), evaluate a termination condition for the processing of thecurrent interval X, wherein the termination condition is TRUE if anumber of conditions are satisfied, including if f^(I)(a) contains zeroand if f^(I)(b) contains zero, and if the termination condition is TRUE,to terminate the processing of the current interval X, and to record Xas a final bound.
 26. The apparatus of claim 25, wherein whileevaluating f(a) to produce the interval result f^(I)(a), the intervalNewton mechanism is additionally configured to perform an intervalNewton step wherein the point of expansion is a if f^(I)(a) does notcontain zero.
 27. The apparatus of claim 25, wherein while evaluatingf(b) to produce the interval result f^(I)(b) the interval Newtonmechanism is additionally configured to perform an interval Newton stepwherein the point of expansion is b if f^(I)(b) does not contain zero.28. The apparatus of claim 25, wherein if f^(I)(x) contains zero andf′(X) contains zero, the termination condition for processing thecurrent interval X is TRUE if f^(I)(a) contains zero, f^(I)(b) containszero, f^(I)(x₁) contains zero and f^(I)(x₂) contains zero; wherein x₁ isthe midpoint between a and x; and wherein x₂ is the midpoint between xand b.
 29. The apparatus of claim 28, wherein if f^(I)(x) contains zeroand if f′(X) contains zero, and if either f^(I)(x₁) or f^(I)(x₂) doesnot contain zero, the interval Newton mechanism is additionallyconfigured to: split the interval X in half; and to apply the intervalNewton method to each half separately.
 30. The apparatus of claim 25,wherein if f^(I)(x) does not contain zero and if f′(X) contains zero,the interval Newton mechanism is additionally configured to terminatethe processing of the current interval X, and to record X as a finalbound on condition that: the width of the interval X divided by themagnitude of the interval X is less than a first threshold value; andthe magnitude of f(X) is less than a second threshold value.
 31. Theapparatus of claim 25, wherein if a given Newton step does not reducethe width of an interval by at least half, the interval Newton mechanismis additionally configured to split the interval in half, and to applythe interval Newton method to each of the two halves separately.
 32. Theapparatus if claim 25, wherein if an interval Newton step results in twointervals, the interval Newton mechanism is additionally configured toapply the interval Newton method to each of the two intervalsseparately.
 33. The apparatus of claim 25, wherein if the result of aninterval Newton step is the empty interval, the interval Newtonmechanism is additionally configured to return to process anotherinterval.
 34. The apparatus of claim 25, wherein if f^(I)(a) containszero and if f^(I)(b) contains zero, the interval Newton mechanism isadditionally configured to return to process another interval.
 35. Theapparatus of claim 25, wherein if f^(I)(a) does not contain zero or iff^(I)(b) does not contain zero, the interval Newton mechanism isadditionally configured to perform an interval Newton step wherein thepoint of expansion is the midpoint of the interval X; and wherein if theresult of the interval Newton step about the midpoint is the emptyinterval, the interval Newton mechanism is additionally configured toreturn to process another interval.
 36. The apparatus of claim 35,wherein the interval Newton mechanism is additionally configured toterminate the processing of the current interval X after a predeterminednumber of iterations.